Well-Posedness for a System of Integro-Differential Equations
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Well-Posedness for a System of Integro-Differential Equations Irene Benedetti1
· Luca Bisconti2
© Foundation for Scientific Research and Technological Innovation 2017
Abstract We give sufficient conditions for the existence, the uniqueness and the continuous dependence on initial data of the solution to a system of integro-differential equations with superlinear growth on the nonlinear term. As possible applications of our methods we consider two epidemic models: a perturbed versions of the well-known integro-differential Kendall SIR model, and a SIRS-like model. Keywords Integro-differential equations · Nonlocal evolution equations · Spatial SIRS models Mathematics Subject Classification Primary: 45J05 · 47G20; Secondary: 45G10
Introduction In this paper we consider the evolution of a vector field u = u(x, t) ∈ Rn , (x, t) ∈ R × R+ 0, depending on space and time and satisfying the following equation ut = [w ⊗ (B1 (k ∗ u))] B2 u − B3 u,
(1)
where w ∈ Rn is assigned, Bi ,i = 1, 2, 3 are n ×n-matrices, and k : R → R+ 0 is a probability density. Here, (k ∗ u)(x, t) = R k(x − y)u(y, t)dy, (x, t) ∈ R × R+ denotes the convolution 0 product of k and u componentwise.
B
Irene Benedetti [email protected] Luca Bisconti [email protected]
1
Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2
Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy
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Differ Equ Dyn Syst
Equations of type (1) appear in population dynamics and epidemiological modelling: for instance, when n = 3, (1) can be assimilated to the case of SIRS endemic models in one spatial dimension (see, e.g., [1,22] for some recent papers on this subject) which are generalizations of the standard SIRS model [18] (see also [13,28]). The original SIR model was introduced in 1927 by Kermack and McKendrick in [20]. This mathematical model takes into account a population which admits only three possible compartments: susceptible (S), infected (I), and removed (R), where these densities are only time dependent, i.e., S = S(t), I = I (t) and R = R(t), t ≥ 0. The SIRS model is an extension of the SIR model that allows recovered members to be free of infection and rejoin the susceptible class. In [25,26] a model in which contacts between individuals are spatially distributed (as for spores spread from a crop rust or virus particles spread by a sneeze) is considered. More precisely, at each location x individuals leave the susceptible class for the infective class at a rate proportional to the product of a weighted average of the number of infective individuals and the density of susceptibles at point x and time t, where k(x − y) is the density function for the proportion of infectives at y that contact susceptibles at x, depending only on the distance between x and y. Moreover, the complexity of the model increases if there exist n different strains of the virus, each of which attacks the population separately and is governed by its ow
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